Question

Calculate the image distance and describe the image orientation for an object placed 12.0 cm from a convex mirror with a focal length of 18.0 cm.

Answer 1

To calculate the image distance for an object placed in front of a convex mirror, we can use the mirror formula:

\[
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
\]

Where:
- \( f \) is the focal length of the mirror (positive for convex mirrors),
- \( d_o \) is the object distance (positive, as the object is in front of the mirror),
- \( d_i \) is the image distance (positive for virtual images formed by convex mirrors, measured from the mirror).

Given:
- \( f = +18.0 \, \text{cm} \) (since it is a convex mirror, the focal length is considered positive),
- \( d_o = 12.0 \, \text{cm} \).

Now we can rearrange the mirror formula to find \( d_i \):

\[
\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}
\]

Substituting in the values:

\[
\frac{1}{d_i} = \frac{1}{18.0} - \frac{1}{12.0}
\]

Calculating the right side:

First, find a common denominator, which for 18 and 12 is 36.

\[
\frac{1}{18.0} = \frac{2}{36}
\]
\[
\frac{1}{12.0} = \frac{3}{36}
\]

Now substitute these into the equation:

\[
\frac{1}{d_i} = \frac{2}{36} - \frac{3}{36} = \frac{-1}{36}
\]

Taking the inverse gives us the image distance:

\[
d_i = -36 \, \text{cm}
\]

The negative sign indicates that the image is virtual and located behind the mirror.

### Image Orientation

For a convex mirror:
- The image is always virtual (indicated by the negative image distance).
- The image is upright (not inverted).
- The image is smaller than the object.

In summary:
- The image distance \( d_i \) is \( -36 \, \text{cm} \).
- The image is virtual, upright, and smaller than the object.